Quaternion-Kähler symmetric space

In convex analysis, Danskin's theorem is a theorem which provides information about the derivatives of a function of the form

$f(x)=\max _{z\in Z}\phi (x,z).$

The theorem has applications in optimization, where it sometimes is used to solve minimax problems.

Statement

The theorem applies to the following situation. Suppose $\phi (x,z)$ is a continuous function of two arguments,

$\phi :{\mathbb {R} }^{n}\times Z\rightarrow {\mathbb {R} }$

where $Z\subset {\mathbb {R} }^{m}$ is a compact set. Further assume that $\phi (x,z)$ is convex in $x$ for every $z\in Z$.

Under these conditions, Danskin's theorem provides conclusions regarding the differentiability of the function

$f(x)=\max _{z\in Z}\phi (x,z).$

To state these results, we define the set of maximizing points $Z_{0}(x)$ as

$Z_{0}(x)=\left\{{\overline {z}}:\phi (x,{\overline {z}})=\max _{z\in Z}\phi (x,z)\right\}.$

Danskin's theorem then provides the following results.

Convexity

$f(x)$ is convex.

Directional derivatives

The directional derivative of $f(x)$ in the direction $y$, denoted $D_{y}\ f(x)$, is given by

$D_{y}f(x)=\max _{z\in Z_{0}(x)}\phi '(x,z;y),$

where $\phi '(x,z;y)$ is the directional derivative of the function $\phi (\cdot ,z)$ at $x$ in the direction $y$.

Derivative

$f(x)$ is differentiable at $x$ if $Z_{0}(x)$ consists of a single element ${\overline {z}}$. In this case, the derivative of $f(x)$ (or the gradient of $f(x)$ if $x$ is a vector) is given by

${\frac {\partial f}{\partial x}}={\frac {\partial \phi (x,{\overline {z}})}{\partial x}}.$

Subdifferential

If $\phi (x,z)$ is differentiable with respect to $x$ for all $z\in Z$, and if $\partial \phi /\partial x$ is continuous with respect to $z$ for all $x$, then the subdifferential of $f(x)$ is given by

$\partial f(x)=\mathrm {conv} \left\{{\frac {\partial \phi (x,z)}{\partial x}}:z\in Z_{0}(x)\right\}$

where $\mathrm {conv}$ indicates the convex hull operation.

See also

• Maximum theorem
• Envelope theorem
• Hotellings Lemma

References

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